Page # Early inflation, Hot Big Bang, Decelerating Expansion, Late inflation by Dark Energy. Lecture 14. Early Inflation

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1 Early inflaion, Ho Big Bang, Deeleraing Expansion, Lae inflaion by Dark Energy Leure 14 Early Inflaion A possible soluion for: Why a ho Big Bang? Horizon problem Flaness problem Monopole problem Origin of primordial sruure 198: Inflaion Alan Guh) Early Inflaion: he firs 1- seonds Universe born from nohing? A quanum fluuaion produes a iny bubble of False auum. High vauum energy drives exponenial expansion, also known as inflaion. Universe expands by huge faor in iny fraion of seond, as false vauum reurns o rue vauum. Expansion so fas ha virual parile-aniparile pairs ge separaed o beome real pariles and anipariles. Srehes ou all sruures, giving a fla geomery and uniform T and ρ, wih iny ripples. Inflaion launhes he Ho Big Bang! Plank Unis Plank Lengh: E M h λ λ 1/ λ RS ) h M RS G M h/ G 1/ ~ LP ~ 1 5 m Plank Time h/ G 1/ L P P 5 ~ 1 4 s Plank Mass MP Quanum auum Heisenberg Unerainy priniple: de Broglie wavelengh ~ Shwarzshild radius ΔE Δ ~ h/ Can violae energy onservaion: Borrow energy from he vauum, bu only for a shor ime. To reae a parile-aniparile pair, need ΔE > m. These virual pairs live only briefly, Δ ~ h / ΔE. 1/ Plank Energy h/ LP h/ ~ 1 5 m p ~ 119 Ge/ G LP G h/ 5 1/ 19 EP MP ~ 1 Ge G Limis of Quanum Mehanis and General Relaiviy. Need Quanum Graviy heory as ye unknown) o desribe physis a hese sales. γ γ e+ _ q e- q auum is no empy. Filled wih all ypes of virual pairs.

2 Like waves on he sea. Quanum auum Quanum fields osillae in many possible wave modes x, ) ψ i x, ) A x, ),. Eah wave mode is a Harmoni Osillaor, Poenial energy:, ψ i, A, ) m + m ψ ψ + Ladder of disree Energy Saes: Zero-poin energy: En) n + 1 h / ω E) 1 h / ω h λ n n 1 n Quanum vauum all wave modes in ground sae. E n number of pariles n 1.. ε va auum Energy Densiy dn k) E k) d x d k d k h / k g π ) 4π k dk k dk g h / 16π g h / 4π Waves in a Box: L n λ k π λ π n Δk π L L dn d x d k g g L x L y L z Δk x Δk y Δk z π k 4 4 max k min ) Shor waves dominae. Densiy of saes in 6-D phase spae: Eah mode is a simple harmoni osillaor. ) 1 E k ) Phoons have g polarisaions. Elerons have g spin saes. Zero-poin energy per mode: h h / k λ Cosmologial Consan Problem Range of wavelenghs in he Cosmologial box: λ > L p ~ h / G 1/ ~ 1 5 m k max ~ π λ < ~ 4 Mp ~ 1 6 m k min ~ π H /H ε va g h / k 4 4 π max k min ) g h / 4 π π g h / 16π 4 ~ E P Ω va ρ va ~ π g h / ρ 4 ri L 8π G ~ 8π g /H p H L ~ 1 1 p Cosmologial Consan problem: Observe Ω Λ ~.7 Predi Ω Λ ~ 1 1 Why is Ω Λ so small, ye no exaly zero? Why does quanum vauum energy no graviae? Flaness Problem Why was he iniial geomery fla? If exaly fla, hen always so. To be roughly fla oday, mus iniially be inredibly lose o fla: 8π Gρ Ωx) H x 4 + Ω M x + Ω Λ x 4 + Ω M x + Ω Λ + 1 Ω ) x Ωx ) 1 ε x ~ ε ~ 1 5 x ~ /H ~ 1 61 Ω 1+ ε x Flaness problem: Why was Ω iniially so lose o 1? How did he Universe know preisely how fas o expand? Horizon Problem Why is he universe almos perfely isoropi? Disan regions were never in ausal ona. Ye we see: Same CMB emperaure o 1 par in 1 5. Similar galaxy disribuions. Thousands of galaxies, jus a few sars. HDF Norh The Hubble Deep Fields Similar galaxy disribuions, supporing he Cosmologial Priniple. HDF Souh

3 WMAP all-sky T.7 K dt / T ~ 1-5 The Horizon: How far an we see? Dipole and foreground galaxy removed suppose R) R ) α 1 ) 1 if α < 1 d α d 1 α χ H η ln if α 1 ) 1 R 1 α R α 1 1 R) 1 ) 1 if α > 1 α 1 1 α η As 1 >, Finie for α < 1, Infinie for α 1, >1. maer - dominaed : α / radiaion - dominaed : α 1/ χh Snapsho a z11 of quanum fluuaions srehed by inflaion. χh Dark maer poenial wells ha seed laer galaxy formaion. L H R ) R R ) d R) 1+ z ) dx Hx) 1+z x z 1+ z ) H ΩM x ) H z ) H z dx We don see any. 1/ kp h ΩM H Why is he universe devoid of magnei monopoles? Expands by faor 1 + z 11 o ~ Mp oday. Phase Transiions among Fores Quanum Graviy η elero-weak E EW ~ 5Ge χh srong weak eleromagnei E GUT ~ 115 Ge χh GUT E P ~ 119 Ge How did hese, ausally disonneed regions know wha emperaure o be, o 1 par in 15? The GUT Grand Unified Theory) phase ransiion should produe opologial defes ha look like magnei monopoles. x 4 ΩR + x ΩM Ω Ω x M 5 M. ΩR Magnei Monopole Problem Parile Horizon a z 11 R graviy Today: k T ~ 1.6 e Inflaion launhes he Big Bang During inflaion: Parile horizon expands R ) d H 1 ) R e L H ) e R) H 1 Dramai ooling 1 1 x 1+ z e H T T 1+ z) e H R R Geomery flaens R eh H R 1/? expand by faor f e H in ime ln f H Wha ame before? Early inflaion replaes he Big Bang singulariy. Launhes he Ho Big Bang. 8π G ρ va 1/ H ~ Horizon problem: Expands a ausally-onneed region. Flaness problem: Flaens he geomery. Monopole problem: Moves primordial monopoles beyond horizon. Seeding sruures: Srehes ou small quanum fluuaions. 8π Gρ ΩR x 4 + ΩM x + ΩΛ H ΩR x 4 + ΩM x + ΩΛ + 1 Ω ) x ΩΛ 1 Ω ) 1 x ΩΛ + 1 Ω ) x ΩΛ Ωx)

4 How muh inflaion needed? Presen horizon: R H ~ / H ~ 4 Mp ~ 1 6 m Plank ime : P ~ 1 4 s pos - inflaion redshif : z P ~ k T CMB f z P ~ f 1 m horizon ~ P ~ ~ 1 5 m E P ~ 119 Ge 1.6 e ~ 1 GUT phase ransiion : E GUT ~ 1 15 Ge E ~ kt ~ R -1 ~ -1/ GUT ~ E P E GUT P ~1 8 P ~ 1 5 s redshif ~ z GUT ~ E GUT ~ 115 Ge k T CMB 1.6 e ~ 18 L GUT L GUT f z GUT ~ f 1 9 m Need inflaion faor f ~1 9 ~ e 67 horizon ~ GUT ~ 1 8 ~ 1 7 m Inflaion faor f ~1 17 ~ e 4 Time ~ ln f ) / H ~ 4 GUT ~ 1 - s Requiremens for Inflaion: Friedmann momenum equaion: R R 4π G ε + p) H w Ω w x 1+w) 1+ w Aeleraion requires a suffiienly negaive pressure: p < - ε / w < -1 / Inflaion requires he dominan omponen o have w < -1 /. Dark Energy is driving inflaion now: w Λ 1 p Λ ε Λ R R + 8π G ε H Λ Ω Λ R exp H Λ ) H Λ H Ω Λ Λ Bu oday s Dark Energy is negligible a early imes. Wha auses Early Inflaion? An evolving Salar Field? Harmoni poenial Higgs poenial During inflaion: Salar Field Dynamis Jus afer inflaion: Today: Klein-Gordon equaion wave equaion for a massive salar field. E p + m 4 E i h / p i h / For a spaially-uniform field: d d m massive field : M P P ) m 4 m h / M P P ) Higgs field : a + b 4 auum energy sars large, delines o a lae imes. Kinei energy of he osillaions is damped. Re-heas he Universe, reaing all ypes of parileaniparile pairs, launhing he Ho Big Bang. Sponaneous Symmery Breaking Why sar in he false vauum, no near ~? Before GUT phase ransiion: Afer GUT phase ransiion: false vauum rue vauum Tunnel o rue vauum? No. Laen hea released, onvers o parile-aniparile pairs, re-heaing he universe. Small exess 1-9 ) of pariles. Salar Field Equaion of Sae Equaion of Sae: w p ϕ h / p / ε ) ) ε ϕ h / +1 w -1/ -1 << h ) + ) >> h Uniform field: ) << Inflaion makes spaial gradiens small. ) Required for inflaion: w p ε h / + h / < 1 ε + p h / < h / <1 The poenial energy mus dominae he kinei energy.

5 Salar Field Dynamis Equaion of Sae: p w p / ε h / ) ε h / + ) Evoluion of energy densiy: ε x 1+w) d dx /x H) ε dε dx dx d 1+ w) ε x H) H ε + p) x Evoluion of uniform salar field: ε h / + Hε + p) H h / + H h / Salar Field Dynamis Hubble Drag: Aeleraion damped by expansion: + H h / Inflaion makes spaial gradiens small. ) Slow-Roll Approximaion: H h / Friedmann equaion: H 8π G ε 8π G Terminal veloiy: h / H Required for Inflaion: h / h / 9H h / 5 4π G Like a snowflake falling / <<1 Inflaion requires a very fla poenial. Long Slow Roll Inflaion Ending in a Ho Big Bang. Long slow roll aross he plaeau poenial dominaed) gives > 6 e-foldings of inflaion ooling,flaening). Laen hea released during kinei energy dominaed) rapid roll and damped osillaions a he end fills he universe wih phoons and parile-aniparile pairs, launhing he Ho Big Bang. Iniial quanum vauum fluuaions mean differen regions finish a slighly differen imes, giving he small 1-5 ) emperaure/densiy ripples we see on he CMB. A Muli-verse? Chaoi Inflaion? auum wih quanum fluuaions. Many bubbles of false vauum inflae, wih: Differen physial onsans. Differen spaial dimensions. ANTHROPIC PRINCIPLE: Mos bubbles quikly re-ollapse or expand oo fas o form sars or are oherwise unsuiable as habias for Life. Is our visible universe par of a big bubble lasing long enough, and wih suiable physial laws, o allow beings like us o evolve?